Export animation from blender to a threes.js program
There are two things here: 


you need help with three.js exporter addon and
the thing you are trying to export isn't supported.


The current Three.js exporter will export certain types of animation, but  an animation where the Boolean modifier is of prime importance (modifying the mesh every frame) isn't going to work as you hoped. For that to work Three.js would need to have a way to deal with the boolean operation, and it would need to be analogous to Blender's Boolean modifier. To my knowledge Three.js has no such feature.

The exporter by MrDoob and associates strips out modifiers by applying them if possible. The boolean modifier can't really be applied per frame unless you export the mesh (per frame) to a format that Three.js knows how to handle (per frame). But that turns into a different question, that you might want to ask on github issue tracker of Three.js